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\begin{document}

\keywords{Dirichlet $L$-function, generalized Riemann conjecture, nontrivial zeros,
Goldbach's conjecture, Riemann zeta function}

\subjclass[2020]{11M06, 11M26}

\title[Generalized Riemann Conjecture]{On the distribution of the nontrivial zeros for the Dirichlet $L$-functions}

\author{Xiao-Jun Yang$^{1,2,3}$}

\email{dyangxiaojun@163.com; xjyang@cumt.edu.cn}

\address{$^{1}$ School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China}
\address{$^{2}$ State Key Laboratory for Geomechanics and Deep Underground Engineering,
China University of Mining and Technology, Xuzhou 221116, China}
\address{$^{3}$ School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China}


\begin{abstract}
This paper addresses a variant of the product for the Dirichlet $L$--functions.
We propose a completely detailed proof for the truth of the
generalized Riemann conjecture for the Dirichlet $L$--functions, which
states that the real part of the nontrivial zeros is $1/2$.
The Wang and Hardy--Littlewood theorems are also discussed
with removing the need for it. The results are applicable
to show the truth of the Goldbach's conjecture.
\end{abstract}

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\section{Introduction} {\label{sec:1}}

The Dirichlet $L$--function $L\left( {s,\chi } \right)$ was formulated by
German mathematician Johann Peter Gustav Lejeune Dirichlet in 1837 for the
meromorphic continuation of the function defined by the series~\cite{1}
\begin{equation}
\label{eq1}
L\left( {s,\chi } \right)=\sum\limits_{n=1}^\infty {\chi \left( n
\right)s^{-n}} ,
\end{equation}
where $\chi \left( n \right)$~is a Dirichlet character~(mod $q>1)$,
$Re\left( s \right)>1$, $n\in \mathds{N}$ and $s\in \mathds{C}$. Here, $q$ is the prime, $\mathds{N}$
and $\mathds{C}$ are the sets of the natural numbers and complex numbers, and
$Re\left( s \right)=\sigma \in \mathds{R}$ and $Im\left( s \right)=t\in \mathds{
R}$ are the real and imaginary parts of the complex variable $s=\sigma
\mbox{+}it\in \mathds{C}$, where $i=\sqrt {-1} $.

For $Re\left( s \right)>1$ the Euler product representation for Eq.~(\ref{eq1}) can
be expressed in the form~\cite{2}:
\begin{equation}
\label{eq2}
L\left( {s,\chi } \right)=\prod\limits_p {\left( {1-\chi \left( p
\right)p^{-s}} \right)^{-1}} .
\end{equation}
Let the primitive character $\chi ^\ast $ with the modulus $q>1_{ }$ such
that~\cite{3}
\begin{equation}
\label{eq3}
\chi \left( n \right)=\left\{ {\begin{array}{l}
 \chi ^\ast \left( n \right)\mbox{ }if\mbox{ gcd}\left( {n,q} \right)=1, \\
 0\mbox{ }if\mbox{ gcd}\left( {n,q} \right)\ne 1. \\
 \end{array}} \right.
\end{equation}
By Eq.~(\ref{eq3}) and $Re\left( s \right)>1$, we rewrite Eq.~(\ref{eq1}) as~\cite{3}:
\begin{equation}
\label{eq4}
L\left( {s,\chi } \right)=L\left( {s,\chi ^\ast }
\right)\prod\limits_{p\left| q \right.} {\left( {1-\frac{\chi ^\ast \left( p
\right)}{p^s}} \right)} ,
\end{equation}
where
\begin{equation}
\label{eq5}
L\left( {s,\chi ^\ast } \right)=\prod\limits_p {\left( {1-\chi ^\ast \left(
p \right)p^{-s}} \right)^{-1}} .
\end{equation}
Let $\chi _0 $ be the principal character with the modulus~ $q>1$ such that~\cite{2,3}
\begin{equation}
\label{eq6}
L\left( {s,\chi _0 } \right)=\zeta \left( s \right)\prod\limits_{p\left| q
\right.} {\left( {1-\frac{1}{p^s}} \right)} \mbox{ }\left( {Re\left( s
\right)>1} \right),
\end{equation}
where the Riemann zeta function is denoted by~\cite{4}
\begin{equation}
\label{eq7}
\zeta \left( s \right)=\sum\limits_{n=1}^\infty {s^{-n}} \mbox{ }\left(
{Re\left( s \right)>1} \right).
\end{equation}
Let $\chi ^\ast $ be a primitive character modulo $q$ with $q>1$. Then the
product for $L\left( {s,\chi } \right)$ can be represented as follows~\cite{3}:
\begin{equation}
\label{eq8}
L\left( {s,\chi ^\ast } \right)=\left( {\frac{q}{\pi }}
\right)^{-\frac{s+\hbar \left( {\chi ^\ast } \right)}{2}}\cdot \frac{\xi
\left( {s,\chi ^\ast } \right)}{\Gamma \left( {\frac{s+\hbar \left( {\chi
^\ast } \right)}{2}} \right)}
\end{equation}
subject to
\begin{equation}
\label{eq9}
\hbar \left( {\chi ^\ast } \right)=\left\{ {\begin{array}{l}
 \mbox{0 if }\chi \left( {-1} \right)=1, \\
 \mbox{1 if }\chi \left( {-1} \right)=-1, \\
 \end{array}} \right.
\end{equation}
and
\begin{equation}
\label{eq10}
\xi \left( {s,\chi ^\ast } \right)=\xi \left( {0,\chi ^\ast } \right)\cdot
e^{B\left( {\chi ^\ast } \right)s}\cdot \prod\limits_{k=1}^\infty {\left(
{1-\frac{s}{\rho _k }} \right)} e^{s/\rho _k }
\end{equation}
where $s\in \mathds{C}$, $\xi \left( {0,\chi ^\ast } \right)\ne 0$, $B\left(
{\chi ^\ast } \right)=\xi ^{\left( 1 \right)}/\xi \left( {0,\chi ^\ast }
\right)$, $\Gamma $ is the gamma function~\cite{5}, and $\rho _k $ run all zeros
of $\xi \left( {s,\chi ^\ast } \right)$.

As the above results, we may present the followings~\cite{3}:

\textbf{Cases 1.}
For $q=1$,
\begin{equation}
\label{eq11}
L\left( {s,\chi } \right)=\zeta \left( s \right)\mbox{ }\left( {Re\left( s
\right)>1} \right);
\end{equation}

\textbf{Cases 2.}
For $\chi =\chi _0 $ with the modulus~$q>1$,
\begin{equation}
\label{eq12}
L\left( {s,\chi } \right)=L\left( {s,\chi _0 } \right)=\zeta \left( s
\right)\prod\limits_{p\left| q \right.} {\left( {1-\frac{1}{p^s}} \right)}
\mbox{ }\left( {Re\left( s \right)>1} \right);
\end{equation}

\textbf{Cases 3.}
Let $\chi ^\ast $ be a primitive character modulo $q$ with
$q>1$. Then
\begin{equation}
\label{eq13}
L\left( {s,\chi } \right)=L\left( {s,\chi ^\ast } \right)=\xi \left( {0,\chi
^\ast } \right)\cdot \left( {\frac{q}{\pi }} \right)^{-\frac{s+\hbar \left(
{\chi ^\ast } \right)}{2}}\cdot e^{B\left( {\chi ^\ast } \right)s}\cdot
\frac{\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{\rho _k }} \right)}
e^{s/\rho _k }}{\Gamma \left( {\frac{s+\hbar \left( {\chi ^\ast }
\right)}{2}} \right)}.
\end{equation}

For \textbf{Cases 1} it is well known that Eq.~(\ref{eq11}) has a pole at $s=1$ with
residue$_{ }1$~\cite{6}, the trivial zeros $s=-2h$ with $h\in \mathds{N}$~\cite{7},
and the nontrivial zeros $s_k =1/2+i\varpi _k $ for $k\in \mathds{N}$
(proved proved by author in the different methods
~\cite{8,9}), where $\varpi _k $ are the Riemann-Siegel zeros, confirmed by the
Riemann-Siegel formula~\cite{10}. Let $\mathds{Z}$ be the set of the integral
numbers. For \textbf{Cases 2} it is seen that Eq.~(\ref{eq12}) has the zeros $s=2\pi
ij/\left( {\log p} \right)$ with $p\left| q \right.$ and $j\in \mathds{Z}$ (see~\cite{11},
p.460) and a pole at $s=1$ with residues $\upsilon \left( q \right)/q$
(see~\cite{2}, p.334), where $\upsilon \left( q \right)$ is the Euler's totient
function. For \textbf{Cases 3} it is shown that Eq.~(\ref{eq13}) has simple zeros
$s=-\hbar \left( {\chi ^\ast } \right)-2h$ with $h\in \mathds{N}$ (see~\cite{11},
p.460;~\cite{2} p.333) and the nontrivial zeros $\rho _k =\alpha _k +i\beta _k $,
where $\beta _k $ are the Siegel zeros, confirmed by the Siegel formula
\cite{12}. Let $N\left( {T,\chi } \right)$ denote the number of zeros of Eq.
~(\ref{eq13}) for $Re\left( s \right)=1/2$, $T>2$ and $\left| {Im\left( s \right)}
\right|\le T$ such that~\cite{13}
\begin{equation}
\label{eq14}
N\left( {T,\chi } \right)=\frac{T}{\pi }\log \frac{qT}{2\pi }-\frac{T}{\pi
}+O\left( {\frac{\log \left( {qT} \right)}{\log \log \left( {qT} \right)}}
\right).
\end{equation}
It is clearly seen that the nontrivial zeros of Eq.~(\ref{eq13}) exist and that Eq.~
(\ref{eq13}) has infinitely many nontrivial zeros by using Eq.~(\ref{eq14}).

\begin{conjecture}
\label{C1}

The generalized Riemann conjecture for Eq.~(\ref{eq1}) states
the real part of the nontrivial zeros is $1/2$.
\end{conjecture}

According to Davenport (see~\cite{2}, p.124), \textit{Conjecture \ref{C1}} was formulated in
1884 by Adolf Piltz. There have been applicable to consider the Goldbach's
conjecture by Hardy and Littlewood in 1923~\cite{14,15} and by Wang in 1962
~\cite{16}. Hardy and Littlewood~\cite{17,18,19} proved that if \textit{Conjecture \ref{C1}}
 is true, almost all even numbers are sums of two primes and that
every large odd number is the sum of three primes.

Recently, author have proposed two classes of the Riemann zeta function,
which can be represented by (the first term was by Hadamard~\cite{9,11,20} and
the second term was by author~\cite{9})
\begin{equation}
\label{eq15}
\begin{array}{l}
 \zeta \left( s \right)\\
 =\xi \left( s \right)/\Re \left( s \right) \\
 =\frac{\xi \left( 0 \right)e^{s\Im }}{\left( {s-1} \right)\Gamma \left(
{\frac{s}{2}+1} \right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{s_k }}
\right)e^{s/s_k }} \\
 =\frac{\xi \left( {1/2} \right)e^{s\Im }}{\left( {s-1} \right)\Gamma \left(
{\frac{s}{2}+1} \right)}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s-\frac{1}{2}}{s_k -\frac{1}{2}}} \right)e^{s/s_k }} \\
 =\frac{\xi \left( s \right)e^{s\Im }}{\left( {s-1} \right)\Gamma \left(
{\frac{s}{2}+1} \right)}\prod\limits_{k=1}^\infty {e^{s/s_k }} , \\
 \end{array}
\end{equation}
provided that there exists the entire Riemann zeta function, given as (the
first term was by Hadamard~\cite{9,11,20} and the second term was by author~\cite{9})
\begin{equation}
\label{eq16}
\xi \left( s \right)=\xi \left( 0 \right)e^{s\Im _0
}\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{s_k }} \right)e^{s/s_k }}
=\xi \left( {1/2} \right)e^{s\Im _0 }\prod\limits_{k=1}^\infty {\left(
{1-\frac{s-\frac{1}{2}}{s_k -\frac{1}{2}}} \right)e^{s/s_k }} ,
\end{equation}
where
\begin{equation}
\label{eq17}
\Re \left( s \right)=\left( {s-1} \right)\pi ^{-s/2}\Gamma \left( {s/2+1}
\right),
\end{equation}
\begin{equation}
\label{eq18}
\Im =\log 2\pi -1-\frac{\widetilde{a}}{2},
\end{equation}
and
\begin{equation}
\label{eq19}
\Im _0 =\log 2\pi -1-\frac{\widetilde{a}}{2}-\frac{1}{2}\log \pi ,
\end{equation}
with the Euler's constant $\widetilde{a}$.

As is well known, \textit{Conjecture \ref{C1}} is an unsolved important
mathematical problem in analytic number theory up to now. By inspired by the above results to
structure the variant of the product for the Riemann zeta function, the main
targets of the paper are to propose a variant of the product for Eq.~(\ref{eq13}),
to prove \textit{Conjecture \ref{C1}} for Eq.~(\ref{eq1}) and to present a detailed account
of applications of the Hardy--Littlewood and Wang theorems to the Goldbach's
conjecture directly removing the need for \textit{Conjecture \ref{C1}}. The
structure of the paper is given as follows. In Section \ref{sec:2}, we introduce the
results related to the Riemann zeta function and Dirichlet $L$--functions. In Section \ref{sec:3},
we give the detailed proof of \textit{Conjecture \ref{C1}}. In Section \ref{sec:4} we apply
to obtain the properties via Dirichlet $L$--functions, get the representations for
Wang theorems, and apply the Hardy--Littlewood
theorems to obtain the Goldbach's conjecture. Finally, we draw the
conclusion in Section \ref{sec:5}.

\section{Fundamental results}{\label{sec:2}}

We now consider the variants of the products for the Riemann zeta
function and Dirichlet $L$--functions.

\subsection{A variant of the product for the Riemann zeta function}{\label{sec:2.1}}

We now consider the variant of the product for the Riemann zeta function.

\begin{lemma}
\label{L1}

There exists
\begin{equation}
\label{eq20}
\zeta \left( 1 \right)\ne 0,
\end{equation}
\begin{equation}
\label{eq21}
\zeta \left( 0 \right)=-\frac{1}{2}\ne 0,
\end{equation}
and
\begin{equation}
\label{eq22}
\zeta \left( {\frac{1}{2}} \right)\ne 0.
\end{equation}
\end{lemma}

\begin{proof}
See the results of Landau~\cite{11}.
\end{proof}

\begin{remark}
Eq.~(\ref{eq20}) was proved in 1896 by Hadamard~\cite{21} and
Vallee-Poussin~\cite{22}. Eqs.~(\ref{eq21}) and~(\ref{eq22}) were discussed in Landau~\cite{11}.


It is proved by Riemann~\cite{4} that
\begin{equation}
\label{eq23}
\zeta \left( s \right)=\frac{\xi \left( s \right)}{\Re \left( s \right)},
\end{equation}
where~\cite{11,20}
\begin{equation}
\label{eq24}
\zeta \left( s \right)=\xi \left( s \right)/\Re \left( s \right)=\frac{\xi
\left( 0 \right)e^{s\Im }}{\left( {s-1} \right)\Gamma \left( {\frac{s}{2}+1}
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{s_k }} \right)e^{s/s_k
}}
\end{equation}
and~\cite{8,9}
\begin{equation}
\label{eq25}
\zeta \left( s \right)=\xi \left( s \right)/\Re \left( s \right)=\frac{\xi
\left( {1/2} \right)e^{s\Im }}{\left( {s-1} \right)\Gamma \left(
{\frac{s}{2}+1} \right)}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s-\frac{1}{2}}{s_k -\frac{1}{2}}} \right)e^{s/s_k }} ,
\end{equation}
where
\begin{equation}
\label{eq26}
\Re \left( s \right)=\left( {s-1} \right)\pi ^{-s/2}\Gamma \left( {s/2+1}
\right),
\end{equation}
and
\begin{equation}
\label{eq27}
\Im =\log 2\pi -1-\widetilde{a}/2
\end{equation}
with the Euler's constant $\widetilde{a}$.
\end{remark}

\begin{lemma}
\label{L2}

There exists
\begin{equation}
\label{eq28}
\xi \left( 1 \right)=\xi \left( 0 \right)=\frac{1}{2}\ne 0,
\end{equation}
and
\begin{equation}
\label{eq29}
\xi \left( {\frac{1}{2}} \right)\ne 0.
\end{equation}
\end{lemma}

\begin{proof}
See the results of Broughan (see~\cite{23}, p.49).
\end{proof}

\begin{lemma}
\label{L3}

Let $\Im _0 =\log 2\pi -1-\frac{\widetilde{a}}{2}-\frac{1}{2}\log \pi $,
where $\widetilde{a}$ is the Euler's constant. Then two classes of the
entire Riemann zeta function are equivalent:
\begin{equation}
\label{eq30}
\xi \left( s \right)=\xi \left( 0 \right)e^{s\Im _0
}\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{s_k }} \right)e^{s/s_k }}
\end{equation}
and
\begin{equation}
\label{eq31}
\xi \left( s \right)=\xi \left( {1/2} \right)e^{s\Im _0
}\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{s_k
-\frac{1}{2}}} \right)e^{s/s_k }} ,
\end{equation}
where $s_k $ run the zeros for $\xi \left( s \right)$, $s\in \mathds{C}$ and
$k\in \mathds{N}$.
\end{lemma}

\begin{proof}
According to Hadamard~\cite{20} and Landau~\cite{11}, we have
\begin{equation}
\label{eq32}
\begin{array}{l}
 \xi \left( s \right)\\
 =\xi \left( 0 \right)e^{s\Im _0
}\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{s_k }} \right)e^{s/s_k }}\\
=\xi \left( 0 \right)e^{s\Im _0 }\prod\limits_{k=1}^\infty {\frac{s_k
-s}{s_k }e^{s/s_k }} \\
=\xi \left( 0 \right)e^{s\Im _0
}\prod\limits_{k=1}^\infty {\left( {\frac{s_k -\frac{1}{2}}{s_k
-\frac{1}{2}}\cdot \frac{s_k -s}{s_k }} \right)e^{s/s_k }} \\
=\xi \left( 0 \right)e^{s\Im _0
}\prod\limits_{k=1}^\infty {\frac{s_k -\frac{1}{2}}{s_k }}
\prod\limits_{k=1}^\infty {\left( {\frac{s_k -s}{s_k -\frac{1}{2}}}
\right)e^{s/s_k }} \\
=\xi \left( 0 \right)e^{s\Im _0
}\prod\limits_{k=1}^\infty {\frac{s_k -\frac{1}{2}}{s_k }}
\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{s_k -\frac{1}{2}}}
\right)e^{s/s_k }} \\
 =\xi \left( 0 \right)e^{s\Im _0 }\prod\limits_{k=1}^\infty {\left(
{1-\frac{1}{2s_k }} \right)} \prod\limits_{k=1}^\infty {\left(
{1-\frac{s-\frac{1}{2}}{s_k -\frac{1}{2}}} \right)e^{s/s_k }} \\
=\xi \left( 0
\right)e^{\left( {s-\frac{1}{2}} \right)\Im _0 }\prod\limits_{k=1}^\infty
{\left( {1-\frac{s-\frac{1}{2}}{s_k -\frac{1}{2}}} \right)e^{s/s_k }} \\
 \end{array}
\end{equation}
where
\begin{equation}
\label{eq33}
\xi \left( {\frac{1}{2}} \right)=\xi \left( 0 \right)e^{\frac{\Im _0
}{2}}\prod\limits_{k=1}^\infty {\left( {1-\frac{1}{2s_k }} \right)e^{1/2s_k
}} =\xi \left( 0 \right)\prod\limits_{k=1}^\infty {\left( {1-\frac{1}{2s_k
}} \right)} ,
\end{equation}
which is derived by~\cite{6,11,20}
\begin{equation}
\label{eq34}
\xi \left( s \right)=\xi \left( 0 \right)e^{s\Im _0
}\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{s_k }} \right)e^{s/s_k }}
=\xi \left( 0 \right)\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{s_k }}
\right)}
\end{equation}
and
\begin{equation}
\label{eq35}
e^{\frac{\Im _0 }{2}}\prod\limits_{k=1}^\infty {e^{1/2s_k }} =1.
\end{equation}
Thus, the proof of \textit{Lemma \ref{L3}} is completed.
\end{proof}

\begin{lemma}
\label{L4}

Assume that $s\in \mathds{C}$, $k\in \mathds{N}$, $_{ }$ and $\Im _0 =\log 2\pi
-1-\widetilde{a}/2$, where $\widetilde{a}_{ }$ is the Euler's constant.
Then two classes of the Riemann zeta function are equivalent:
\begin{equation}
\label{eq36}
\zeta \left( s \right)=\frac{\xi \left( 0 \right)e^{s\Im }}{\left( {s-1}
\right)\Gamma \left( {\frac{s}{2}+1} \right)}\prod\limits_{k=1}^\infty
{\left( {1-\frac{s}{s_k }} \right)e^{s/s_k }}
\end{equation}
and~\cite{5,9}
\begin{equation}
\label{eq37}
\zeta \left( s \right)=\frac{\xi \left( {1/2} \right)e^{s\Im }}{\left( {s-1}
\right)\Gamma \left( {\frac{s}{2}+1} \right)}\prod\limits_{k=1}^\infty
{\left( {1-\frac{s-\frac{1}{2}}{s_k -\frac{1}{2}}} \right)e^{s/s_k }} ,
\end{equation}
\end{lemma}

\begin{proof}
By Eq.~(\ref{eq23}), \textit{Lemma \ref{L3}} and $\Re \left( s \right)=\left( {s-1}
\right)\pi ^{-s/2}\Gamma \left( {s/2+1} \right)$, we obtain the required
results.
\end{proof}

\begin{remark}
For the detailed proofs of \textit{Lemmas \ref{L3} and \ref{L4}}, see~\cite{8,9}.

According to Patterson~\cite{23} we have
\begin{equation}
\label{eq38}
s\left( {s-1} \right)\Gamma \left( {\frac{s}{2}} \right)\zeta \left( s
\right)=\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{s_k^\ast }}
\right)} \left( {1-\frac{s}{1-s _k^\ast }} \right)
\end{equation}
such that~\cite{8}
\begin{equation}
\label{eq39}
\xi \left( s \right)=\frac{1}{2}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s}{s_k^\ast }} \right)} \left( {1-\frac{s}{1-s_k^\ast }}
\right)=\xi \left( 0 \right)\prod\limits_{k=1}^\infty {\left(
{1-\frac{s}{s_k^\ast }} \right)} \left( {1-\frac{s}{1-s_k^\ast }}
\right),
\end{equation}
where $s\in \mathds{C}$, $k\in \mathds{N}$, and
\begin{equation}
\label{eq40}
s_k^\ast =Re\left( {s_k } \right)+i\left| {Im\left( {s_k }
\right)} \right|.
\end{equation}
It is proved by author~\cite{8} that for $s\in \mathds{C}$,
\begin{equation}
\label{eq41}
\begin{array}{l}
 \xi \left( s \right)=\xi \left( 0 \right)\prod\limits_{k=1}^\infty {\left(
{1-\frac{s}{s_k^\ast }} \right)} \left( {1-\frac{s}{1-s_k^\ast }}
\right)
=\xi
\left( {\frac{1}{2}} \right)\prod\limits_{k=1}^\infty {\left[
{1-\frac{\left( {s-\frac{1}{2}} \right)^2}{\left( {s_k^\ast
-\frac{1}{2}} \right)^2}} \right]}, \\
 \end{array}
\end{equation}
which implies from Eq.~(\ref{eq23}) that~\cite{8}
\begin{equation}
\label{eq42}
\zeta \left( s \right)=\frac{\xi \left( s \right)}{\Re \left( s
\right)}=\frac{\xi \left( 0 \right)\pi ^{s/2}}{\left( {s-1} \right)\Gamma
\left( {s/2+1} \right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{s
_k^\ast }} \right)} \left( {1-\frac{s}{1-s_k^\ast }} \right),
\end{equation}
\begin{equation}
\label{eq43}
\zeta \left( s \right)=\frac{\xi \left( s \right)}{\Re \left( s
\right)}=\frac{\xi \left( 0 \right)\pi ^{s/2}}{\left( {s-1} \right)\Gamma
\left( {s/2+1} \right)}\prod\limits_{k=1}^\infty {\left[ {1-\frac{s\left(
{1-s} \right)}{s_k^\ast \left( {1-s_k^\ast } \right)}} \right]} ,
\end{equation}
and~\cite{8}
\begin{equation}
\label{eq44}
\zeta \left( s \right)=\frac{\xi \left( s \right)}{\Re \left( s
\right)}=\frac{\xi \left( {\frac{1}{2}} \right)\pi ^{s/2}}{\left( {s-1}
\right)\Gamma \left( {s/2+1} \right)}\prod\limits_{k=1}^\infty {\left[
{1-\frac{\left( {s-\frac{1}{2}} \right)^2}{\left( {s_k^\ast
-\frac{1}{2}} \right)^2}} \right]} ,
\end{equation}
where $s\in \mathds{C}$, $s\ne 1$, and
\begin{equation}
\label{eq45}
\Re \left( s \right)=\left( {s-1} \right)\pi ^{-s/2}\Gamma \left( {s/2+1}
\right).
\end{equation}
Now, from Eq.~(\ref{eq42}), we see that $s _k $, $1-s _k $, $s_k^\ast $ and
$1-s_k^\ast $ are the zeros for $\xi \left( s \right)$ and the
nontrivial zeros for $\zeta \left( s \right)$~\cite{8}. Moreover, $\xi \left( s
\right)$ is a meromorphic continuation to the entire complex plane $s$ with
pole of residue $1$~\cite{6}, and $\xi \left( s \right)$ is an integral function
of order $1$ and simple in the entire complex plane $s$~\cite{8}. By Eqs.~(\ref{eq31}) and
~(\ref{eq44}), it is proved that $Re\left( {s _k } \right)=1/2$, in other words
that the Riemann conjecture is true (for the detailed proof, see~\cite{8,9}).
\end{remark}

\subsection{A variant of the product for the Dirichlet $L$--functions}{\label{sec:2.2}}

In this part we present the variant of the product for the
Dirichlet $L$--functions. We now denote the primitive character modulo $q$ with
$q>1$ by $\chi ^\ast =\chi ^\ast \left( n \right)$.

\begin{lemma}
\label{L5}

If $\chi ^\ast $ is a primitive character modulo $q$ with $q>1$ and $\chi
^\ast \left( n \right)\ne \chi _0 $, then
\begin{equation}
\label{eq46}
L\left( {1,\chi ^\ast } \right)\ne 0,
\end{equation}
\begin{equation}
\label{eq47}
L\left( {0,\chi ^\ast } \right)\ne 0
\end{equation}
and
\begin{equation}
\label{eq48}
L\left( {\frac{1}{2},\chi ^\ast } \right)\ne 0.
\end{equation}
\end{lemma}

\begin{proof}
For the detailed proof of Eq.~(\ref{eq46}), see the work of Dirichlet~\cite{1,3}. Moreover,
Eqs.~(\ref{eq47}) and~(\ref{eq48}) were proved in~\cite{25}.
\end{proof}

\begin{lemma}
\label{L6}

If $\chi ^\ast $ is a primitive character modulo $q$ with $q>1$, then we
have that
\begin{equation}
\label{eq49}
\xi \left( {s,\chi ^\ast } \right)=L\left( {s,\chi ^\ast } \right)\cdot
\left( {\frac{q}{\pi }} \right)^{\frac{s+\hbar \left( {\chi ^\ast }
\right)}{2}}\cdot \Gamma \left( {\frac{s+\hbar \left( {\chi ^\ast }
\right)}{2}} \right),
\end{equation}
is an entire function of order $1$, where $s\in \mathds{C}$ and $\hbar \left(
{\chi ^\ast } \right)$ is defined in~(\ref{eq9}).
\end{lemma}

\begin{proof}
For the detailed proof of \textit{Lemma \ref{L6}}, see~\cite{3}.
\end{proof}

\begin{lemma}
\label{L7}

If $\chi ^\ast $ is a primitive character modulo $q$ with $q>1$, then we
have
\begin{equation}
\label{eq50}
\xi \left( {1,\chi ^\ast } \right)\ne 0,
\end{equation}
\begin{equation}
\label{eq51}
\xi \left( {0,\chi ^\ast } \right)\ne 0
\end{equation}
and
\begin{equation}
\label{eq52}
\xi \left( {\frac{1}{2},\chi ^\ast } \right)\ne 0.
\end{equation}
\end{lemma}

\begin{proof}
For the detailed proofs of Eqs.~(\ref{eq50}),~(\ref{eq51}) and~(\ref{eq52}), see~\cite{11}.
\end{proof}

\begin{lemma}
\label{L8}

Suppose that $\chi ^\ast $ is a primitive character modulo $q$ with $q>1$ and
$\rho _k $ run the nontrivial zeros of $L\left( {s,\chi ^\ast } \right)$.
Then there is
\begin{equation}
\label{eq53}
\xi \left( {s,\chi ^\ast } \right)=\xi \left( {0,\chi ^\ast }
\right)e^{B\left( {\chi ^\ast } \right)s}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s}{\rho _k }} \right)} e^{s/\rho _k },
\end{equation}
where $s\in \mathds{C}$ and
\begin{equation}
\label{eq54}
B\left( {\chi ^\ast } \right)=\frac{\xi ^{\left( 1 \right)}}{\xi }\left(
{0,\chi ^\ast } \right).
\end{equation}
\end{lemma}

\begin{proof}
For the detailed proof of \textit{Lemma \ref{L8}}, see~\cite{3}.
\end{proof}

\begin{lemma}
\label{L9}

Suppose that $\chi ^\ast $ is a primitive character modulo $q$ with $q>1$ and
$\rho _k $ run the nontrivial zeros of $L\left( {s,\chi ^\ast } \right)$.
Then there is
\begin{equation}
\label{eq55}
\frac{\xi ^{\left( 1 \right)}}{\xi }\left( {s,\chi ^\ast } \right)=B\left(
{\chi ^\ast } \right)+\sum\limits_{k=1}^\infty {\left( {\frac{1}{s-\rho _k
}+\frac{1}{\rho _k }} \right)} .
\end{equation}
\end{lemma}

\begin{proof}
For the detailed proof of \textit{Lemma \ref{L9}}, see~\cite{3}.
\end{proof}

\begin{lemma}
\label{L10}

Suppose that $\chi ^\ast $ is a primitive character modulo $q$ with $q>1$ and
$\rho _k $ run the nontrivial zeros of $L\left( {s,\chi ^\ast } \right)$.
Then we have for $s\in \mathds{C}$,
\begin{equation}
\label{eq56}
L\left( {s,\chi ^\ast } \right)=B\left( {\chi ^\ast } \right)\Phi \left(
{s,\hbar \left( {\chi ^\ast } \right)} \right)L\left( {1-s,\overline {\chi
^\ast } } \right),
\end{equation}
and
\begin{equation}
\label{eq57}
\xi \left( {s,\chi ^\ast } \right)=\varepsilon \left( {\chi ^\ast }
\right)\xi \left( {1-s,\overline {\chi ^\ast } } \right),
\end{equation}
provided that
\begin{equation}
\label{eq58}
\Phi \left( {s,\hbar \left( {\chi ^\ast } \right)} \right)=2^s\pi
^{1/2-s}\Gamma \left( {1-s} \right)\sin \frac{\pi }{2}\left( {s+\hbar \left(
{\chi ^\ast } \right)} \right),
\end{equation}
where $\hbar \left( {\chi ^\ast } \right)$ is defined in~(\ref{eq9}),
\begin{equation}
\label{eq59}
\varepsilon \left( {\chi ^\ast } \right)=\frac{\tau \left( {\chi ^\ast }
\right)}{i^{\hbar \left( {\chi ^\ast } \right)}\sqrt q },
\end{equation}
the Gauss sum $\tau \left( {\chi ^\ast } \right)$ of $\chi ^\ast $ is
denoted by
\begin{equation}
\label{eq60}
\tau \left( {\chi ^\ast } \right)=\sum\limits_{\theta =1}^q {\chi ^\ast
\left( \theta \right)e^{\theta /q}} ,
\end{equation}
and $\overline {\chi ^\ast } _{ }$ is the complex conjugate character
to $\chi ^\ast $.
\end{lemma}

\begin{proof}
For the detailed proof of \textit{Lemma \ref{L10}}, see~\cite{2,3}.
\end{proof}

\begin{lemma}
\label{L11}

Suppose that $\chi ^\ast $ is a primitive character modulo $q$ with $q>1$.
Then we have
\begin{equation}
\label{eq61}
\left| {\tau \left( {\chi ^\ast } \right)} \right|=\sqrt q .
\end{equation}
\end{lemma}

\begin{proof}
For the detailed proof of \textit{Lemma \ref{L11}}, see~\cite{3}.
\end{proof}

Thus, we see that
\begin{equation}
\label{eq62}
\tau \left( {\chi ^\ast } \right)\ne 0
\end{equation}
if $\chi ^\ast $ is a primitive character modulo $q$ with $q>1$.

\begin{theorem}
\label{T1}

Suppose that $\chi ^\ast $ is a primitive character modulo $q$ with $q>1$ and
$\rho _k $ run the nontrivial zeros of $L\left( {s,\chi ^\ast } \right)_{
}$ with $k\in \mathds{N}$. Then there exist the equivalent representations:
\begin{equation}
\label{eq63}
\xi \left( {s,\chi ^\ast } \right)=\xi \left( {0,\chi ^\ast }
\right)e^{B\left( {\chi ^\ast } \right)s}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s}{\rho _k }} \right)} e^{s/\rho _k },
\end{equation}
and
\begin{equation}
\label{eq64}
\xi \left( {s,\chi ^\ast } \right)=\xi \left( {\frac{1}{2},\chi ^\ast }
\right)e^{\left( {s-\frac{1}{2}} \right)B\left( {\chi ^\ast }
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{\rho _k
-\frac{1}{2}}} \right)e^{\left( {s-1/2} \right)/\rho _k }} ,
\end{equation}
where $s\in \mathds{C}$ and
\begin{equation}
\label{eq65}
B\left( {\chi ^\ast } \right)=\frac{\xi ^{\left( 1 \right)}}{\xi }\left(
{0,\chi ^\ast } \right).
\end{equation}
\end{theorem}

\begin{proof}
From Eqs.~(\ref{eq51}) and~(\ref{eq52}),
\begin{equation}
\label{eq66}
\xi \left( {\frac{1}{2},\chi ^\ast } \right)=\xi \left( {0,\chi ^\ast }
\right)e^{\frac{B\left( {\chi ^\ast } \right)}{2}}\prod\limits_{k=1}^\infty
{\left( {1-\frac{1}{2\rho _k }} \right)} \prod\limits_{k=1}^\infty
{e^{1/\left( {2\rho _k } \right)}} \ne 0.
\end{equation}
By \textit{Lemma \ref{L8}}, we get
\begin{equation}
\label{eq67}
\begin{array}{l}
 \xi \left( {s,\chi ^\ast } \right)\\
 =\xi \left( {0,\chi ^\ast }
\right)e^{B\left( {\chi ^\ast } \right)s}\prod\limits_{k=1}^\infty
{e^{s/\rho _k }} \prod\limits_{k=1}^\infty {\left( {1-\frac{s}{\rho _k }}
\right)} \\
 =\xi \left( {0,\chi ^\ast } \right)e^{B\left( {\chi ^\ast }
\right)s}\prod\limits_{k=1}^\infty {e^{s/\rho _k }}
\prod\limits_{k=1}^\infty {\left( {\frac{\rho _k -s}{\rho _k }} \right)} \\
 =\xi \left( {0,\chi ^\ast } \right)e^{B\left( {\chi ^\ast }
\right)s}\prod\limits_{k=1}^\infty {e^{s/\rho _k }}
\prod\limits_{k=1}^\infty {\left( {\frac{\rho _k -\frac{1}{2}}{\rho _k
-\frac{1}{2}}\cdot \frac{\rho _k -s}{\rho _k }} \right)} \\
 =\xi \left( {0,\chi ^\ast } \right)e^{B\left( {\chi ^\ast }
\right)s}\prod\limits_{k=1}^\infty {e^{s/\rho _k }}
\prod\limits_{k=1}^\infty {\left( {\frac{\rho _k -\frac{1}{2}}{\rho _k
}\cdot \frac{\rho _k -s}{\rho _k -\frac{1}{2}}} \right)} \\
 =\xi \left( {0,\chi ^\ast } \right)e^{B\left( {\chi ^\ast }
\right)s}\prod\limits_{k=1}^\infty {\frac{\rho _k -\frac{1}{2}}{\rho _k }}
\prod\limits_{k=1}^\infty {e^{s/\rho _k }} \prod\limits_{k=1}^\infty {\left(
{\frac{\rho _k -s}{\rho _k -\frac{1}{2}}} \right)} \\
 =\xi \left( {0,\chi ^\ast } \right)e^{B\left( {\chi ^\ast }
\right)s}\prod\limits_{k=1}^\infty {\frac{\rho _k -\frac{1}{2}}{\rho _k }}
\prod\limits_{k=1}^\infty {e^{s/\rho _k }} \prod\limits_{k=1}^\infty {\left[
{\frac{\left( {\rho _k -\frac{1}{2}} \right)-\left( {s-\frac{1}{2}}
\right)}{\rho _k -\frac{1}{2}}} \right]} \\
 =\xi \left( {0,\chi ^\ast } \right)e^{B\left( {\chi ^\ast }
\right)s}\prod\limits_{k=1}^\infty {\frac{\rho _k -\frac{1}{2}}{\rho _k }}
\prod\limits_{k=1}^\infty {e^{s/\rho _k }} \prod\limits_{k=1}^\infty {\left(
{1-\frac{s-\frac{1}{2}}{\rho _k -\frac{1}{2}}} \right)} \\
 =\xi \left( {0,\chi ^\ast } \right)e^{B\left( {\chi ^\ast }
\right)s}\prod\limits_{k=1}^\infty {\frac{\rho _k -\frac{1}{2}}{\rho _k }}
\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{\rho _k
-\frac{1}{2}}} \right)e^{s/\rho _k }} \\
 =\xi \left( {0,\chi ^\ast } \right)e^{sB\left( {\chi ^\ast }
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{1}{2\rho _k }} \right)}
\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{\rho _k
-\frac{1}{2}}} \right)e^{s/\rho _k }} \\
 =\xi \left( {\frac{1}{2},\chi ^\ast } \right)e^{\left( {s-\frac{1}{2}}
\right)B\left( {\chi ^\ast } \right)}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s-\frac{1}{2}}{\rho _k -\frac{1}{2}}} \right)e^{\left( {s-1/2}
\right)/\rho _k }} , \\
 \end{array}
\end{equation}
which is the required result.
\end{proof}

\begin{theorem}
\label{T2}

Suppose that $\chi ^\ast $ is a primitive character modulo $q$ with $q>1$ and
$\rho _k $ run the nontrivial zeros of $L\left( {s,\chi ^\ast } \right)$
with $k\in \mathds{N}$. Then there exist the equivalent representations:
\begin{equation}
\label{eq68}
L\left( {s,\chi ^\ast } \right)=\frac{\xi \left( {0,\chi ^\ast }
\right)\cdot \left( {\frac{q}{\pi }} \right)^{-\frac{s+\hbar \left( {\chi
^\ast } \right)}{2}}\cdot e^{B\left( {\chi ^\ast } \right)s}}{\Gamma \left(
{\frac{s+\hbar \left( {\chi ^\ast } \right)}{2}} \right)}\cdot
\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{\rho _k }} \right)} e^{s/\rho
_k },
\end{equation}
and
\begin{equation}
\label{eq69}
L\left( {s,\chi ^\ast } \right)=\frac{\xi \left( {\frac{1}{2},\chi ^\ast }
\right)\cdot \left( {\frac{q}{\pi }} \right)^{-\frac{s+\hbar \left( {\chi
^\ast } \right)}{2}}\cdot e^{\left( {s-\frac{1}{2}} \right)B\left( {\chi
^\ast } \right)}}{\Gamma \left( {\frac{s+\hbar \left( {\chi ^\ast }
\right)}{2}} \right)}\cdot \prod\limits_{k=1}^\infty {\left(
{1-\frac{s-\frac{1}{2}}{\rho _k -\frac{1}{2}}} \right)e^{\left( {s-1/2}
\right)/\rho _k }} ,
\end{equation}
where $s\in \mathds{C}$ and
\begin{equation}
\label{eq70}
B\left( {\chi ^\ast } \right)=\frac{\xi ^{\left( 1 \right)}}{\xi }\left(
{0,\chi ^\ast } \right).
\end{equation}
\end{theorem}

\begin{proof}
With use of \textit{Lemma \ref{L6}} and \textit{Theorem \ref{T1}}, we present~\cite{11}
\begin{equation}
\label{eq71}
L\left( {s,\chi ^\ast } \right)=\frac{\xi \left( {0,\chi ^\ast }
\right)\cdot \left( {\frac{q}{\pi }} \right)^{-\frac{s+\hbar \left( {\chi
^\ast } \right)}{2}}\cdot e^{B\left( {\chi ^\ast } \right)s}}{\Gamma \left(
{\frac{s+\hbar \left( {\chi ^\ast } \right)}{2}} \right)}\cdot
\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{\rho _k }} \right)} e^{s/\rho
_k },
\end{equation}
and
\begin{equation}
\label{eq72}
L\left( {s,\chi ^\ast } \right)=\frac{\xi \left( {\frac{1}{2},\chi ^\ast }
\right)\cdot \left( {\frac{q}{\pi }} \right)^{-\frac{s+\hbar \left( {\chi
^\ast } \right)}{2}}\cdot e^{\left( {s-\frac{1}{2}} \right)B\left( {\chi
^\ast } \right)}}{\Gamma \left( {\frac{s+\hbar \left( {\chi ^\ast }
\right)}{2}} \right)}\cdot \prod\limits_{k=1}^\infty {\left(
{1-\frac{s-\frac{1}{2}}{\rho _k -\frac{1}{2}}} \right)e^{\left( {s-1/2}
\right)/\rho _k }} ,
\end{equation}
which are the desired results, where $s\in \mathds{C}$.
\end{proof}

\begin{remark}
There exists (see~\cite{2}, p.83)
\begin{equation}
\label{eq73}
B\left( {\chi ^\ast } \right)=\frac{\xi ^{\left( 1 \right)}}{\xi }\left(
{0,\chi ^\ast } \right)=-\frac{\xi ^{\left( 1 \right)}}{\xi }\left( {1,\chi
^\ast } \right)=-B\left( {\overline {\chi ^\ast } }
\right)-\sum\limits_{k=1}^\infty {\left( {\frac{1}{1-\rho _k }+\frac{1}{\rho
_k }} \right)} .
\end{equation}
\end{remark}

\begin{lemma}
\label{L12}
Let $\chi ^\ast $ be a primitive character modulo $q$ with $q>1$.
$L\left( {s,\chi ^\ast } \right)$ has an infinity of zeros $\rho _k $ in the
critical strip $0\le Re\left( s \right)\le 1$.
\end{lemma}

\begin{proof}
See the result of Davenport (see~\cite{2}, p.82).
\end{proof}

\subsection{The Goldbach's problems}{\label{sec:2.3}}

We now introduce the Hardy-Littlewood and Wang's theorems with the need of
\textit{Conjecture \ref{C1}}.

\begin{lemma}
\label{L13}(Hardy-Littlewood Theorem I) ~\cite{14}

If \textit{Conjecture \ref{C1}} is true, then every odd number $m>5$ is the sum of
three primes.
\end{lemma}

\begin{proof}
See the work of Hardy and Littlewood~\cite{14} and the paper of Deshouillers and
coauthors~\cite{15} under the condition of truth of \textit{Conjecture \ref{C1}}.
\textit{Lemma \ref{L13}} was proved in 2013 by Helfgott~\cite{26} without~\textit{Conjecture~\ref{C1}}
and a detailed account
of the numerical verification for \textit{Lemma \ref{L13}} was shown by Helfgott
and Platt in 2013~\cite{27}.
\end{proof}

For the sake of brevity, we denote the following proposition by $\left(
{1,\mathds{X}} \right)$~\cite{16,28}:

Every sufficiently large even integer is a sum of a prime and an almost
prime of at most $\mathds{X}$ prime divisors.

\begin{lemma}
\label{L14}(Wang Theorem I)~\cite{16}

If \textit{Conjecture \ref{C1}} is true, then $\left( {1,4} \right)$ is valid,
where $\delta _2 \ge 3.237/2.237$.
\end{lemma}

\begin{proof}
See the work of Wang~\cite{16}.
\end{proof}

\begin{lemma}
\label{L15}(Wang Theorem II)~\cite{16}

If \textit{Conjecture \ref{C1}} is true, then $\left( {1,3} \right)$ is valid,
where $\delta _1 \ge 2.475/1.475$.
\end{lemma}

\begin{proof}
See the work of Wang~\cite{16}.
\end{proof}

In fact, Chen proved in 1973 and 1978 that $\left( {1,2} \right)$ is true~\cite{28,29}.
The records of verification of the strong Goldbach's conjecture holds to $N_1
=4\times 10\wedge 18$~\cite{30}.

\begin{lemma}
\label{L16} (Hardy-Littlewood Theorem II)~\cite{17}

If \textit{Conjecture \ref{C1}} is true, every even number $m>2$ is sums of two
primes.
\end{lemma}

\begin{proof}
See the work of Hardy and Littlewood~\cite{17}. Based on it, the work of Granville~\cite{18} gives
the detailed proof of \textit{Lemma \ref{L16}} on the condition of truth of
\textit{Conjecture \ref{C1}}.
\end{proof}

\section{A detailed proof for the generalized Riemann conjecture} {\label{sec:3}}
We now apply the variant of the product for the
Dirichlet $L$--functions to present the complete
proof for \textit{Conjecture \ref{C1}}.

\subsection{Family 1: $\chi =\chi _0$ and $q=1$}

When $\chi =\chi _0$ and $q=1$, we have~\cite{3}
\begin{equation}
\label{eq74}
\zeta \left( s \right)=\sum\limits_{n=1}^\infty {s^{-n}} \mbox{ }\left(
{Re\left( s \right)>1} \right).
\end{equation}
According to \textit{Lemma \ref{L3}}, we have the representation
\begin{equation}
\label{eq75}
\begin{array}{l}
 \zeta \left( s \right)\\
 =\frac{\xi \left( 0 \right)e^{s\Im }}{\left( {s-1}
\right)\Gamma \left( {\frac{s}{2}+1} \right)}\prod\limits_{k=1}^\infty
{\left( {1-\frac{s}{s_k }} \right)e^{s/s_k }} \\
 =\frac{\xi \left( {1/2} \right)e^{s\Im }}{\left( {s-1} \right)\Gamma \left(
{\frac{s}{2}+1} \right)}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s-\frac{1}{2}}{s_k -\frac{1}{2}}} \right)e^{s/s_k }} \\
 =\frac{\xi \left( s \right)e^{s\Im }}{\left( {s-1} \right)\Gamma \left(
{\frac{s}{2}+1} \right)}\prod\limits_{k=1}^\infty {e^{s/s_k }} \\
 \end{array}
\end{equation}
where $s\in \mathds{C}$ and $s\ne 1$, and so we have
\begin{equation}
\label{eq76}
\xi \left( s \right)=\xi \left( 0 \right)\prod\limits_{k=1}^\infty {\left(
{1-\frac{s}{s_k }} \right)} =\xi \left( {1/2}
\right)\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{s_k
-\frac{1}{2}}} \right)}
\end{equation}
and
\begin{equation}
\label{eq77}
\Xi \left( \beta \right)=\xi \left( {\frac{1}{2}+i\beta } \right)=\xi \left(
0 \right)\prod\limits_{k=1}^\infty {\left( {1-\frac{\frac{1}{2}+i\beta }{s_k
}} \right)} =\xi \left( {1/2} \right)\prod\limits_{k=1}^\infty {\left(
{1-\frac{i\beta }{s_k -\frac{1}{2}}} \right)} ,
\end{equation}
where $\xi \left( 0 \right)\ne 0$ and $\xi \left( {1/2} \right)\ne 0$ (for
the details, see \textit{Lemma \ref{L2}}).

It is well knows that Eqs.~(\ref{eq76}) and~(\ref{eq77}) are the entire functions of order
$1$.

By Eqs.~(\ref{eq75}) and~(\ref{eq76}), we have
\begin{equation}
\label{eq78}
\zeta \left( s \right)=0
\end{equation}
such that
\begin{equation}
\label{eq79}
s-1\ne 0,
\end{equation}
\begin{equation}
\label{eq80}
\xi \left( s \right)=0
\end{equation}
and
\begin{equation}
\label{eq81}
\frac{1}{\Gamma \left( {\frac{s}{2}+1} \right)}=0.
\end{equation}
With use of Eq.~(\ref{eq77}), we have~\cite{8,9}
\begin{equation}
\label{eq82}
\Xi \left( {\varpi _k } \right)=\xi \left( {1/2}
\right)\prod\limits_{k=1}^\infty {\left( {1-\frac{i\varpi _k }{s_k
-\frac{1}{2}}} \right)} =0.
\end{equation}
Since Eq.~(\ref{eq76}) is an entire function of order $1$, it follows that
\begin{equation}
\label{eq83}
1-\frac{i\varpi _k }{s_k -\frac{1}{2}}=0,
\end{equation}
where $k\in \mathds{N}$.

From Eq.~(\ref{eq83}), we have
\begin{equation}
\label{eq84}
s_k =1/2+i\varpi _k
\end{equation}
and~\cite{8,9}
\begin{equation}
\label{eq85}
\widetilde{s_k }=\frac{1}{2}\pm i\left| {\varpi _k } \right|,
\end{equation}
where $k\in \mathds{N}$, $i=\sqrt {-1} $, and $\varpi _k $ are the
Riemann-Siegel zeros.

Putting Eq.~(\ref{eq76}) into Eq.~(\ref{eq84}), we get
\begin{equation}
\label{eq86}
\xi \left( s \right)=\xi \left( {1/2} \right)\prod\limits_{k=1}^\infty
{\left( {1-\frac{s-\frac{1}{2}}{i\varpi _k }} \right)} .
\end{equation}
Substituting $s_k =\omega _k +i\varpi _k $ into Eq.~(\ref{eq87}), we get
\begin{equation}
\label{eq87}
\xi \left( {\omega _k +i\varpi _k } \right)=\xi \left( {1/2}
\right)\prod\limits_{k=1}^\infty {\left( {1-\frac{\omega _k +i\varpi _k
-\frac{1}{2}}{i\varpi _k }} \right)} =0.
\end{equation}
Since Eq.~(\ref{eq77}) is an entire function of order $1$, it follows that
\begin{equation}
\label{eq88}
1-\frac{\omega _k +i\varpi _k -\frac{1}{2}}{i\varpi _k }=0,
\end{equation}
which leads to
\begin{equation}
\label{eq89}
\omega _k =1/2,
\end{equation}
or, alternatively,
\begin{equation}
\label{eq90}
s_k =1/2+i\varpi _k .
\end{equation}
With use of Eq.~(\ref{eq79}), Eq.~(\ref{eq75}) has a pole at $s=1$ with residue $1$
~\cite{6}. By Eq.~(\ref{eq81}), Eq.~(\ref{eq75}) has the trivial zeros $s=-2h$ with $h\in
\mathds{N}$~\cite{6}. In view of Eq.~(\ref{eq89}), we find that Eq.~(\ref{eq75}) has the
nontrivial zeros $s_k =1/2+i\varpi _k $~\cite{8,9}.

\subsection{Family 2: $\chi =\chi _0$ and $q>1$}

When $\chi =\chi _0$ and $q>1$, we get~\cite{3}
\begin{equation}
\label{eq91}
L\left( {s,\chi _0 } \right)=\zeta \left( s \right)\prod\limits_{p\left| q
\right.} {\left( {1-\frac{1}{p^s}} \right)} \mbox{ }\left( {Re\left( s
\right)>1} \right).
\end{equation}
By \textit{Lemmas \ref{L3}},~\ref{L4},~(\ref{eq75}) and~(\ref{eq91}), we show that
\begin{equation}
\label{eq92}
\begin{array}{l}
 L\left( {s,\chi _0 } \right)\\
 =\zeta \left( s \right)\prod\limits_{p\left| q
\right.} {\left( {1-\frac{1}{p^s}} \right)} \mbox{ } \\
 =\frac{\xi \left( 0 \right)e^{s\Im }\mbox{ }}{\left( {s-1} \right)\Gamma
\left( {\frac{s}{2}+1} \right)}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s}{s_k }} \right)e^{s/s_k }} \prod\limits_{p\left| q \right.}
{\left( {1-\frac{1}{p^s}} \right)} \\
 =\frac{\xi \left( {1/2} \right)e^{s\Im }}{\left( {s-1} \right)\Gamma \left(
{\frac{s}{2}+1} \right)}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s-\frac{1}{2}}{s_k -\frac{1}{2}}} \right)e^{s/s_k }}
\prod\limits_{p\left| q \right.} {\left( {1-\frac{1}{p^s}} \right)} \mbox{ }
\\
 =\frac{\xi \left( s \right)e^{s\Im }}{\left( {s-1} \right)\Gamma \left(
{\frac{s}{2}+1} \right)}\prod\limits_{p\left| q \right.} {\left(
{1-\frac{1}{p^s}} \right)} \prod\limits_{k=1}^\infty {e^{s/s_k }} \mbox{ ,}
\\
 \end{array}
\end{equation}
where $s\in \mathds{C}$ and $s\ne 1$.

By Eq.~(\ref{eq92}) and \textit{Lemma \ref{L2}}, we have
\begin{equation}
\label{eq93}
L\left( {s,\chi _0 } \right)=0
\end{equation}
such that
\begin{equation}
\label{eq94}
1-\frac{1}{p^s}=0
\end{equation}
and
\begin{equation}
\label{eq95}
\zeta \left( s \right)=0.
\end{equation}
By \textbf{Family 1}, it is clearly seen that Eq.~(\ref{eq95}) has the nontrivial
zeros $s_k =1/2+i\varpi _k $ for $k\in \mathds{N}$~\cite{8,9} and the trivial zeros
$s=-2h$ with $h\in \mathds{N}$~\cite{6}, and a pole at $s=1$ with residue
$\upsilon \left( q \right)/q=\prod\limits_{p\left| q \right.} {\left(
{1-\frac{1}{p}} \right)} $~(\cite{3}, p.334), where $\upsilon \left( q \right)$ is
the Euler's totient function. From Eq.~(\ref{eq94}) it is shown that Eq.~(\ref{eq93})
has the pure imaginary zeros $s=2\pi ij/\left( {\log p} \right)$ with
$p\left| q \right.$ and $j\in \mathds{Z}$ (see~\cite{11}, p.460).

\subsection{Family 3: $\chi =\chi ^\ast$ and $\chi ^\ast \left( {-1}
\right)=1$}

When $\chi \ne \chi _0$ and $\chi
\left( {-1} \right)=1$, we write~\cite{3}
\begin{equation}
\label{eq96}
\xi \left( {s,\chi ^\ast } \right)=L\left( {s,\chi ^\ast } \right)\Gamma
\left( {\frac{s+k\left( {\chi ^\ast } \right)}{2}} \right)\left(
{\frac{q}{\pi }} \right)^{\frac{s+k\left( {\chi ^\ast } \right)}{2}}
\end{equation}
as
\begin{equation}
\label{eq97}
\xi \left( {s,\chi ^\ast } \right)=L\left( {s,\chi ^\ast } \right)\Gamma
\left( {\frac{s}{2}} \right)\left( {\frac{q}{\pi }} \right)^{\frac{s}{2}}
\end{equation}
with
\begin{equation}
\label{eq98}
\xi \left( {s,\chi ^\ast } \right)=\varepsilon \left( {\chi ^\ast }
\right)\xi \left( {1-s,\overline {\chi ^\ast } } \right),
\end{equation}
where $s\in \mathds{C}$,
\begin{equation}
\label{eq99}
k\left( {\chi ^\ast } \right)=0,
\end{equation}
\begin{equation}
\label{eq100}
\varepsilon \left( {\chi ^\ast } \right)=\frac{\tau \left( {\chi ^\ast }
\right)}{\sqrt q },
\end{equation}
and the Gauss sum of $\chi ^\ast $ by
\begin{equation}
\label{eq101}
\tau \left( {\chi ^\ast } \right)=\sum\limits_{\theta =1}^q {\chi ^\ast
\left( \theta \right)e^{\theta /q}} .
\end{equation}
Since $\xi \left( {s,\chi ^\ast } \right)$ is an integral function of order
$1$, we have
\begin{equation}
\label{eq102}
\xi \left( {s,\chi ^\ast } \right)=\xi \left( {0,\chi ^\ast }
\right)e^{B\left( {\chi ^\ast } \right)s}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s}{\rho _k }} \right)} e^{s/\rho _k }
\end{equation}
such that (see \textit{Theorem \ref{T1}})
\begin{equation}
\label{eq103}
\begin{array}{l}
 \xi \left( {s,\chi ^\ast } \right)\\
 =\xi \left( {0,\chi ^\ast }
\right)e^{B\left( {\chi ^\ast } \right)s}\prod\limits_{k=1}^\infty
{e^{s/\rho _k }} \prod\limits_{k=1}^\infty {\left( {1-\frac{s}{\rho _k }}
\right)} \\
 =\xi \left( {0,\chi } \right)e^{B\left( \chi
\right)s}\prod\limits_{k=1}^\infty {\left( {1-\frac{1}{2\rho _k }} \right)}
\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{\rho _k
-\frac{1}{2}}} \right)e^{s/\rho _k }} . \\
 \end{array}
\end{equation}
By Eq.~(\ref{eq102}), we have
\begin{equation}
\label{eq104}
\xi \left( {\frac{1}{2},\chi ^\ast } \right)=\xi \left( {0,\chi ^\ast }
\right)e^{\frac{B\left( {\chi ^\ast } \right)}{2}}\prod\limits_{k=1}^\infty
{\left( {1-\frac{1}{2\rho _k }} \right)} \prod\limits_{k=1}^\infty
{e^{1/\left( {2\rho _k } \right)}}
\end{equation}
such that
\begin{equation}
\label{eq105}
\frac{\xi \left( {s,\chi ^\ast } \right)}{\xi \left( {\frac{1}{2},\chi ^\ast
} \right)}=\frac{\xi \left( {0,\chi ^\ast } \right)e^{B\left( {\chi ^\ast }
\right)s}\prod\limits_{k=1}^\infty {\left( {1-\frac{1}{2\rho _k }} \right)}
\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{\rho _k
-\frac{1}{2}}} \right)e^{s/\rho _k }} }{\xi \left( {0,\chi ^\ast }
\right)e^{\frac{B\left( {\chi ^\ast } \right)}{2}}\prod\limits_{k=1}^\infty
{\left( {1-\frac{1}{2\rho _k }} \right)} \prod\limits_{k=1}^\infty
{e^{1/\left( {2\rho _k } \right)}} },
\end{equation}
where (see \textit{Lemma \ref{L7}} for the details)
\begin{equation}
\label{eq106}
\xi \left( {0,\chi ^\ast } \right)\ne 0
\end{equation}
and
\begin{equation}
\label{eq107}
\xi \left( {\frac{1}{2},\chi ^\ast } \right)\ne 0.
\end{equation}
By \textit{Theorem \ref{T1}}, we arrive at
\begin{equation}
\label{eq108}
\xi \left( {s,\chi ^\ast } \right)=\xi \left( {\frac{1}{2},\chi ^\ast }
\right)e^{B\left( {\chi ^\ast } \right)\left( {s-\frac{1}{2}}
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{\rho _k
-\frac{1}{2}}} \right)e^{\left( {s-1/2} \right)/\rho _k }} .
\end{equation}
From~(\ref{eq97}),~(\ref{eq108}) and \textit{Theorem \ref{T2}}, we suggest that
\begin{equation}
\label{eq109}
\begin{array}{l}
 L\left( {s,\chi ^\ast } \right)\\
 =\left( {\frac{q}{\pi }}
\right)^{-\frac{s}{2}}\frac{\xi \left( {s,\chi ^\ast } \right)}{\Gamma
\left( {\frac{s}{2}} \right)} \\
 =\frac{\left( {\frac{q}{\pi }} \right)^{-\frac{s}{2}}\cdot \xi \left(
{\frac{1}{2},\chi ^\ast } \right)\cdot e^{B\left( {\chi ^\ast }
\right)\left( {s-\frac{1}{2}} \right)}}{\Gamma \left( {\frac{s}{2}}
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{\rho _k
-\frac{1}{2}}} \right)e^{\left( {s-1/2} \right)/\rho _k }} . \\
 \end{array}
\end{equation}
Since Eq.~(\ref{eq108}) is an integral function of order $1$, Eq.~(\ref{eq109}) is an
integral function of order $1$.

Substituting $s=1/2+i\beta $ into Eq.~(\ref{eq108}), we have~\cite{11}
\begin{equation}
\label{eq110}
\Xi \left( {\beta ,\chi ^\ast } \right)=\xi \left( {1/2+i\beta ,\chi ^\ast }
\right)=\xi \left( {\frac{1}{2},\chi ^\ast } \right)e^{i\beta B\left( {\chi
^\ast } \right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{i\beta }{\rho _k
-\frac{1}{2}}} \right)e^{i\beta /\rho _k }} .
\end{equation}
It is shown that Eq.~(\ref{eq66}) is an integral function of order $1$.

Taking $\beta =\beta _k $, we have from Eq.~(\ref{eq66}) that
\begin{equation}
\label{eq111}
\Xi \left( {\beta _k ,\chi ^\ast } \right)=\xi \left( {\frac{1}{2},\chi
^\ast } \right)e^{i\beta _k B\left( {\chi ^\ast }
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{i\beta _k }{\rho _k
-\frac{1}{2}}} \right)e^{i\beta _k /\rho _k }} =0.
\end{equation}
By Eq.~(\ref{eq67}) and \textit{Lemma \ref{L7}}, we may arrive at
\begin{equation}
\label{eq112}
1-\frac{i\beta _k }{\rho _k -\frac{1}{2}}=0,
\end{equation}
and by Eq.~(\ref{eq112}) we get
\begin{equation}
\label{eq113}
\rho _k =\frac{1}{2}+i\beta _k .
\end{equation}
By Eq.~(\ref{eq113}), Eqs.~(\ref{eq108}),~(\ref{eq109}) and~(\ref{eq110}) can be rewritten as
\begin{equation}
\label{eq114}
\xi \left( {s,\chi ^\ast } \right)=\xi \left( {\frac{1}{2},\chi ^\ast }
\right)e^{B\left( {\chi ^\ast } \right)\left( {s-\frac{1}{2}}
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{i\beta _k
}} \right)e^{\left( {s-1/2} \right)/\left( {1/2+i\beta _k } \right)}} ,
\end{equation}
\begin{equation}
\label{eq115}
L\left( {s,\chi ^\ast } \right)=\frac{\left( {\frac{q}{\pi }}
\right)^{-\frac{s}{2}}\xi \left( {\frac{1}{2},\chi ^\ast } \right)e^{B\left(
{\chi ^\ast } \right)\left( {s-\frac{1}{2}} \right)}}{\Gamma \left(
{\frac{s}{2}} \right)}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s-\frac{1}{2}}{i\beta _k }} \right)e^{\left( {s-1/2} \right)/\left(
{1/2+i\beta _k } \right)}} ,
\end{equation}
and
\begin{equation}
\label{eq116}
\Xi \left( {\beta ,\chi ^\ast } \right)=\xi \left( {1/2+i\beta ,\chi ^\ast }
\right)=\xi \left( {\frac{1}{2},\chi ^\ast } \right)e^{i\beta B\left( {\chi
^\ast } \right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{\beta }{\beta _k
}} \right)e^{i\beta /\left( {1/2+i\beta _k } \right)}} ,
\end{equation}
where $s\in \mathds{C}$.

From Eq.~(\ref{eq60}) and \textit{Lemma \ref{L6}}, we observe that
\begin{equation}
\label{eq117}
L\left( {s,\chi ^\ast } \right)=0,
\end{equation}
which leads to
\begin{equation}
\label{eq118}
\frac{1}{\Gamma \left( {\frac{s}{2}} \right)}=0
\end{equation}
and
\begin{equation}
\label{eq119}
\xi \left( {s,\chi ^\ast } \right)=\xi \left( {\frac{1}{2},\chi ^\ast }
\right)e^{B\left( {\chi ^\ast } \right)\left( {s-\frac{1}{2}}
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{i\beta _k
}} \right)e^{\left( {s-1/2} \right)/\left( {1/2+i\beta _k } \right)}} =0.
\end{equation}
Making use of Eq.~(\ref{eq118}), we find that~\cite{2,3,11}
\begin{equation}
\label{eq120}
s=-2k,
\end{equation}
which are the trivial zeros of Eq.~(\ref{eq109}), where $k\in \mathds{N}\cup \left\{ 0
\right\}$.

Inserting $\rho _k =\alpha _k +i\beta _k $ into Eq.~(\ref{eq119}) implies that
\begin{equation}
\label{eq121}
\begin{array}{l}
 \xi \left( {\alpha _k +i\beta _k ,\chi ^\ast } \right) \\
 =\xi \left( {\frac{1}{2},\chi ^\ast } \right)e^{B\left( {\chi ^\ast }
\right)\left( {\alpha _k +i\beta _k -\frac{1}{2}}
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{\alpha _k +i\beta _k
-\frac{1}{2}}{i\beta _k }} \right)e^{\left( {\alpha _k +i\beta _k -1/2}
\right)/\left( {1/2+i\beta _k } \right)}} \\
 =0. \\
 \end{array},
\end{equation}
With Eq.~(\ref{eq121}) and \textit{Lemma \ref{L6}}, we arrive at
\begin{equation}
\label{eq122}
1-\frac{\alpha _k +i\beta _k -\frac{1}{2}}{i\beta _k }=0,
\end{equation}
and we get
\begin{equation}
\label{eq123}
\alpha _k =\frac{1}{2}.
\end{equation}
Thus,
\begin{equation}
\label{eq124}
\rho _k =1/2+i\beta _k
\end{equation}
or, alternatively,
\begin{equation}
\label{eq125}
Re\left( {\rho _k } \right)=1/2.
\end{equation}

\subsection{Family 4: $\chi =\chi ^\ast$ and $\chi ^\ast \left( {-1}
\right)=-1$}

When $\chi =\chi ^\ast _{ }$ and $\chi
^\ast \left( {-1} \right)=-1_{ }$, we write~\cite{3}
\begin{equation}
\label{eq126}
\xi \left( {s,\chi ^\ast } \right)=L\left( {s,\chi ^\ast } \right)\Gamma
\left( {\frac{s+k\left( {\chi ^\ast } \right)}{2}} \right)\left(
{\frac{q}{\pi }} \right)^{\frac{s+k\left( {\chi ^\ast } \right)}{2}}
\end{equation}
as
\begin{equation}
\label{eq127}
\xi \left( {s,\chi ^\ast } \right)=L\left( {s,\chi ^\ast } \right)\Gamma
\left( {\frac{s+1}{2}} \right)\left( {\frac{q}{\pi }}
\right)^{\frac{s+1}{2}}
\end{equation}
with
\begin{equation}
\label{eq128}
\xi \left( {s,\chi ^\ast } \right)=\varepsilon \left( {\chi ^\ast }
\right)\xi \left( {1-s,\overline {\chi ^\ast } } \right),
\end{equation}
where $s\in \mathds{C}$,
\begin{equation}
\label{eq129}
k\left( {\chi ^\ast } \right)=1,
\end{equation}
\begin{equation}
\label{eq130}
\varepsilon \left( {\chi ^\ast } \right)=\frac{\tau \left( {\chi ^\ast }
\right)}{\sqrt q },
\end{equation}
and the Gauss sum of $\chi ^\ast $ by~\cite{3}
\begin{equation}
\label{eq131}
\tau \left( {\chi ^\ast } \right)=\sum\limits_{\theta =1}^q {\chi ^\ast
\left( \theta \right)e^{\theta /q}} .
\end{equation}
Since $\xi \left( {s,\chi ^\ast } \right)$ is an integral function of order
$1$, by \textit{Theorem \ref{T1}}, we have from Eqs.~(\ref{eq102}) and (\ref{eq108}) that
\begin{equation}
\label{eq132}
\begin{array}{l}
 \xi \left( {s,\chi ^\ast } \right) \\
 =\xi \left( {0,\chi ^\ast } \right)e^{B\left( {\chi ^\ast }
\right)s}\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{\rho _k }} \right)}
e^{s/\rho _k } \\
 =\xi \left( {\frac{1}{2},\chi ^\ast } \right)e^{B\left( {\chi ^\ast }
\right)\left( {s-\frac{1}{2}} \right)}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s-\frac{1}{2}}{\rho _k -\frac{1}{2}}} \right)e^{\left( {s-1/2}
\right)/\rho _k }} . \\
 \end{array}
\end{equation}
Since Eq.~(\ref{eq132}) is an integral function of order $1$, $L\left( {s,\chi
^\ast } \right)$ is an integral function of order $1$.

Taking $s=1/2+i\beta _{ }$ into Eq.~(\ref{eq132}), we get
\begin{equation}
\label{eq133}
\begin{array}{l}
 \Xi \left( {\beta ,\chi ^\ast } \right) \\
 =\xi \left( {1/2+i\beta ,\chi ^\ast } \right) \\
 =\xi \left( {0,\chi ^\ast } \right)e^{B\left( {\chi ^\ast }
\right)s}\prod\limits_{k=1}^\infty {\left( {1-\frac{\frac{1}{2}+i\beta
}{\rho _k }} \right)} e^{s/\rho _k } \\
 =\xi \left( {\frac{1}{2},\chi ^\ast } \right)e^{i\beta B\left( {\chi ^\ast
} \right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{i\beta }{\rho _k
-\frac{1}{2}}} \right)e^{i\beta /\rho _k }} . \\
 \end{array}
\end{equation}
We may find that Eq.~(\ref{eq133}) is an integral function of order $1$.

Substituting $\beta =\beta _k$ into Eq.~(\ref{eq133}), we have
\begin{equation}
\label{eq134}
\Xi \left( {\beta _k ,\chi ^\ast } \right)=\xi \left( {\frac{1}{2},\chi
^\ast } \right)e^{i\beta _k B\left( {\chi ^\ast }
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{i\beta _k }{\rho _k
-\frac{1}{2}}} \right)e^{i\beta _k /\rho _k }} =0.
\end{equation}
By the virtue of Eq.~(\ref{eq134}) and \textit{Lemma \ref{L6}}, we give
\begin{equation}
\label{eq135}
1-\frac{i\beta _k }{\rho _k -\frac{1}{2}}=0,
\end{equation}
and by Eq.~(\ref{eq135}), we arrive at
\begin{equation}
\label{eq136}
\rho _k =\frac{1}{2}+i\beta _k .
\end{equation}
Combining Eqs.~(\ref{eq127}) and~(\ref{eq132}) and~(\ref{eq136}) and using \textit{Theorem \ref{T2}}, we have
\begin{equation}
\label{eq137}
\begin{array}{l}
 L\left( {s,\chi ^\ast } \right)\\
 =\left( {\frac{q}{\pi }}
\right)^{-\frac{s+1}{2}}\frac{\xi \left( {s,\chi ^\ast } \right)}{\Gamma
\left( {\frac{s+1}{2}} \right)} \\
 =\frac{\xi \left( {\frac{1}{2},\chi ^\ast } \right)\cdot \left(
{\frac{q}{\pi }} \right)^{-\frac{s+1}{2}}\cdot e^{B\left( {\chi ^\ast }
\right)\left( {s-\frac{1}{2}} \right)}}{\Gamma \left( {\frac{s+1}{2}}
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{i\beta _k
}} \right)e^{\left( {s-1/2} \right)/\left( {1/2+i\beta _k } \right)}} . \\
 \end{array}
\end{equation}
In view of Eq.~(\ref{eq137}), we obtain
\begin{equation}
\label{eq138}
\frac{1}{\Gamma \left( {\frac{s+1}{2}} \right)}=0
\end{equation}
and
\begin{equation}
\label{eq139}
\xi \left( {s,\chi ^\ast } \right)=\xi \left( {\frac{1}{2},\chi ^\ast }
\right)\cdot e^{B\left( {\chi ^\ast } \right)\left( {s-\frac{1}{2}}
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{i\beta _k
}} \right)e^{\left( {s-1/2} \right)/\left( {1/2+i\beta _k } \right)}} =0.
\end{equation}
By Eq.~(\ref{eq138}), we present~\cite{2,3,11}
\begin{equation}
\label{eq140}
s=-2k-1,
\end{equation}
which is the trivial zeros of Eq.~(\ref{eq137}), where $k\in \mathds{N}\cup \left\{ 0
\right\}$.

Substituting $\rho _k =\alpha _k +i\beta _k $ into Eq.~(\ref{eq139}) we obtain
\begin{equation}
\label{eq141}
\begin{array}{l}
 \xi \left( {\alpha _k +i\beta _k ,\chi ^\ast } \right) \\
 =\xi \left( {\frac{1}{2},\chi ^\ast } \right)\cdot e^{B\left( {\chi ^\ast }
\right)\left( {\alpha _k +i\beta _k -\frac{1}{2}}
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{\alpha _k +i\beta _k
-\frac{1}{2}}{i\beta _k }} \right)e^{\left( {\alpha _k +i\beta _k -1/2}
\right)/\left( {1/2+i\beta _k } \right)}} \\
 =0, \\
 \end{array}
\end{equation}
which implies, by \textit{Lemma \ref{L6}}, that
\begin{equation}
\label{eq142}
1-\frac{\alpha _k +i\beta _k -\frac{1}{2}}{i\beta _k }=0.
\end{equation}
Thus, by Eq.~(\ref{eq123}) we have
\begin{equation}
\label{eq143}
\alpha _k =1/2
\end{equation}
such that
\begin{equation}
\label{eq144}
\rho _k =1/2+i\beta _k .
\end{equation}

By \textit{Lemma \ref{L12}} and using the above results, we clearly see
that the real part of the zeros in the critical trip is
$1/2$ .

It is shown that \textit{Conjecture \ref{C1}} is true.

Hence, we finish the proof of \textit{Conjecture \ref{C1}}.

\begin{remark}

In short, we easily see the followings:
\begin{itemize}
  \item
  When $\chi =\chi _0$ and $q=1$, $L\left( {s,\chi }
\right)=\zeta \left( s \right)$ is extended to be a
meromorphic continuation to the entire complex plane $s$, and has a pole at $s=1$ with residue $_{
}1$, the trivial zeros $s=-2h$ with $h\in \mathds{N}$, and the nontrivial
zeros $s_k =1/2+i\varpi _k $ with $k\in \mathds{N}$,
which lie on the critical line $s=1/2$ and in the critical trip
$0<Re(s)<1$.\\
As shown in Section \ref{sec:1}, \textbf{Case 1} has been proved by authors in \cite{8,9}.
  \item
  When $\chi =\chi _0$ and $q>1$, $L\left( {s,\chi } \right)=L\left(
{s,\chi _0 } \right)$ is extended to be a
meromorphic continuation to the entire complex plane $s$, and has a pole at $s=1$ with residue $\upsilon \left(
q \right)/q=\prod\limits_{p\left| q \right.} {\left( {1-p^{-1}} \right)} $,
the pure imaginary zeros $s=2\pi ij/\left( {\log p} \right)$ with $p\left| q
\right.$ and $j\in \mathds{Z}$, the trivial zeros $s=-2h$ with $h\in \mathds{N}$,
and the nontrivial zeros $s_k =1/2+i\varpi _k $ for $k\in \mathds{N}$.\\
It is seen that \textbf{Cases 2} and \textbf{1} have the same as the
nontrivial zeros $s_k =1/2+i\varpi _k $, a pole at $s=1$ with
different residues, and trivial zeros $s=-2h$ with $h\in \mathds{N}$.
Its nontrivial zeros for $L\left( {s,\chi } \right)=L\left(
{s,\chi _0 } \right)$ lie on the critical line $s=1/2$ and in the critical trip
$0<Re(s)<1$.
\end{itemize}

As shown in \textbf{Case 3} of Section \ref{sec:1}, we have followings:
\begin{itemize}
  \item
  When $\chi =\chi ^\ast$ and $\chi ^\ast \left( {-1} \right)=1$,
\[
L\left( {s,\chi } \right)=L\left( {s,\chi ^\ast } \right)=\left( {q/\pi }
\right)^{-\frac{s}{2}}\xi \left( {s,\chi ^\ast } \right)/\Gamma \left( {s/2}
\right)
\]
is an integral function of order $1$, and has the zeros (all zeros are the nontrivial zeros)
$\rho _k =1/2+i\beta _k $ with $k\in \mathds{N}$.\\
  \item
  When $\chi =\chi ^\ast$ and $\chi ^\ast \left( {-1}
\right)=-1$,
\[
L\left( {s,\chi } \right)=L\left( {s,\chi ^\ast }
\right)=\left( {q/\pi } \right)^{-\frac{s+1}{2}}\xi \left( {s,\chi }
\right)/\Gamma \left( {s/2+1/2} \right)
\]
is an integral function of order
$1$, and has the zeros (all zeros are the nontrivial zeros) $\rho _k=1/2+i\beta _k $ with $k\in \mathds{N}$.
It is observed that in \textbf{Case 3} of Section \ref{sec:1}, they have the nontrivial zeros $\rho _k=1/2+i\beta _k $ with $k\in \mathds{N}$, which
lie on the critical line $s=1/2$ and in the critical trip
$0<Re(s)<1$.
\end{itemize}
\end{remark}

\section{New results and applications} {\label{sec:4}}

In this section we report the new formulas associated with the
Dirichlet $L$--functions. Then
we also give the new representations for the Wang theorems.
Main target of the part is
to present the applications of \textit{Conjecture \ref{C1}} and
Hardy--Littlewood theorems to obtain the Goldbach's conjecture.

\subsection{New formulas for the Dirichlet $L$--functions}

We now give the properties for the Dirichlet $L$--functions.

\begin{theorem}
\label{T3}

Suppose that $\chi ^\ast $ is a primitive character modulo $q$ with $q>1$.
Let $\beta _k $ be the Siegel zeros for the Dirichlet $L$--functions with the
primitive character $\chi ^\ast _{ }$ with $k\in \mathds{N}$. Then there exist
the equivalent representations:
\begin{equation}
\label{eq145}
\xi \left( {s,\chi ^\ast } \right)=\xi \left( {0,\chi ^\ast }
\right)e^{sB\left( {\chi ^\ast } \right)}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s}{\frac{1}{2}+i\beta _k }} \right)} e^{s/\left( {1/2+i\beta _k }
\right)},
\end{equation}
and
\begin{equation}
\label{eq146}
\xi \left( {s,\chi ^\ast } \right)=\xi \left( {\frac{1}{2},\chi ^\ast }
\right)e^{\left( {s-\frac{1}{2}} \right)B\left( {\chi ^\ast }
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{i\beta _k
}} \right)e^{\left( {s-1/2} \right)/\left( {1/2+i\beta _k } \right)}} ,
\end{equation}
where $s\in \mathds{C}$, and
\begin{equation}
\label{eq147}
B\left( {\chi ^\ast } \right)=\frac{\xi ^{\left( 1 \right)}}{\xi }\left(
{0,\chi ^\ast } \right).
\end{equation}
\end{theorem}

\begin{proof}
By \textit{Conjecture \ref{C1}} and \textit{Theorem \ref{T1}}, we give the desired results.
\end{proof}

\begin{theorem}
\label{T4}

Suppose that $\chi ^\ast$ is a primitive character modulo $q$ with
$q>1$ and $\beta _k $ run the Siegel zeros for the Dirichlet $L$--functions
with the primitive character $\chi ^\ast $ with $k\in \mathds{N}$. Then there
exist the equivalent representations:
\begin{equation}
\label{eq148}
L\left( {s,\chi ^\ast } \right)=\frac{\xi \left( {0,\chi ^\ast }
\right)\cdot \left( {\frac{q}{\pi }} \right)^{-\frac{s+\hbar \left( {\chi
^\ast } \right)}{2}}\cdot e^{B\left( {\chi ^\ast } \right)s}}{\Gamma \left(
{\frac{s+\hbar \left( {\chi ^\ast } \right)}{2}} \right)}\cdot
\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{\frac{1}{2}+i\beta _k }}
\right)} e^{s/\left( {1/2+i\beta _k } \right)},
\end{equation}
and
\begin{equation}
\label{eq149}
L\left( {s,\chi ^\ast } \right)=\frac{\xi \left( {\frac{1}{2},\chi ^\ast }
\right)\cdot \left( {\frac{q}{\pi }} \right)^{-\frac{s+\hbar \left( {\chi
^\ast } \right)}{2}}\cdot e^{\left( {s-\frac{1}{2}} \right)B\left( {\chi
^\ast } \right)}}{\Gamma \left( {\frac{s+\hbar \left( {\chi ^\ast }
\right)}{2}} \right)}\cdot \prod\limits_{k=1}^\infty {\left(
{1-\frac{s-\frac{1}{2}}{i\beta _k }} \right)e^{\left( {s-1/2} \right)/\left(
{1/2+i\beta _k } \right)}} ,
\end{equation}
where $s\in \mathds{C}$, $\hbar \left( {\chi ^\ast } \right)$ is defined in
Eq.~(\ref{eq9}), and
\begin{equation}
\label{eq150}
B\left( {\chi ^\ast } \right)=\frac{\xi ^{\left( 1 \right)}}{\xi }\left(
{0,\chi ^\ast } \right).
\end{equation}
\end{theorem}

\begin{proof}
By \textit{Conjecture \ref{C1}} and \textit{Theorem \ref{T2}}, we give the desired results.
\end{proof}

\begin{theorem}
\label{T5}

Suppose that $\chi ^\ast$ is a primitive character modulo $q$ with
$q>1$ and $\beta _k $ are the Siegel zeros. Then there exist the equivalent
representations:
\begin{equation}
\label{eq151}
\Xi \left( {\beta ,\chi ^\ast } \right)=\xi \left( {0,\chi ^\ast }
\right)e^{\left( {1/2+i\beta } \right)B\left( {\chi ^\ast }
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{\frac{1}{2}+i\beta
}{\frac{1}{2}+i\beta _k }} \right)} e^{\left( {1/2+i\beta } \right)/\left(
{1/2+i\beta _k } \right)},
\end{equation}
and
\begin{equation}
\label{eq152}
\Xi \left( {\beta ,\chi ^\ast } \right)=\xi \left( {\frac{1}{2},\chi ^\ast }
\right)e^{i\beta B\left( {\chi ^\ast } \right)}\prod\limits_{k=1}^\infty
{\left( {1-\frac{\beta }{\beta _k }} \right)e^{i\beta /\left( {1/2+i\beta _k
} \right)}} ,
\end{equation}
where $\beta \in \mathds{C}$, $\hbar \left( {\chi ^\ast } \right)$ is defined
in Eq.~(\ref{eq9}), and
\begin{equation}
\label{eq153}
B\left( {\chi ^\ast } \right)=\frac{\xi ^{\left( 1 \right)}}{\xi }\left(
{0,\chi ^\ast } \right).
\end{equation}
\end{theorem}

\begin{proof}
By \textit{Theorem \ref{T3}} and \textit{Conjecture \ref{C1}}, we obtain the desired
results.
\end{proof}

We now define the function by
\begin{equation}
\label{eq154}
\xi \left( {s,\chi _0 } \right)=\Re \left( s \right)L\left( {s,\chi _0 }
\right)=\left( {s-1} \right)\pi ^{-s/2}\Gamma \left( {s/2+1} \right)L\left(
{s,\chi _0 } \right),
\end{equation}
where $s\in \mathds{C}$ and $s\ne 1$.


\begin{remark}
Suppose that $\chi _0 $ is the principal character modulo $q$ with $q>1$.
Then we have that
\begin{equation}
\label{eq155}
\xi \left( {s,\chi _0 } \right)=\xi \left( 0 \right)e^{s\Im _0
}\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{s_k }} \right)e^{s/s_k }}
\prod\limits_{p\left| q \right.} {\left( {1-\frac{1}{p^s}} \right)} \mbox{
}
\end{equation}
and
\begin{equation}
\label{eq156}
\xi \left( {s,\chi _0 } \right)=\xi \left( {1/2} \right)e^{s\Im _0
}\prod\limits_{k=1}^\infty {\left( {1-\frac{s-\frac{1}{2}}{s_k
-\frac{1}{2}}} \right)e^{s/s_k }} \prod\limits_{p\left| q \right.} {\left(
{1-\frac{1}{p^s}} \right)} ,
\end{equation}
where $s\in \mathds{C}$ and $s\ne 1$.

Because \textit{Conjecture \ref{C1}} is true, $1-\rho _k $, $\overline {\rho _k } $
and $1-\overline {\rho _k } $ are the nontrivial zeros for the
Dirichlet $L$--functions, which is in agreement with the results of Montgomery
and Vaughan~\cite{3}.

From Eq.~(\ref{eq152}), we have
\begin{equation}
\label{eq157}
\Xi \left( {\beta ,\chi ^\ast } \right)=\xi \left( {\frac{1}{2},\chi ^\ast }
\right)e^{i\beta B\left( {\chi ^\ast } \right)}\prod\limits_{k=1}^\infty
{\left( {1-\frac{\beta }{\beta _k }} \right)e^{i\beta /\left( {1/2+i\beta _k
} \right)}} ,
\end{equation}
and
\begin{equation}
\label{eq158}
\Xi \left( {-\beta ,\chi ^\ast } \right)=\xi \left( {\frac{1}{2},\chi ^\ast
} \right)e^{-i\beta B\left( {\chi ^\ast } \right)}\prod\limits_{k=1}^\infty
{\left( {1+\frac{\beta }{\beta _k }} \right)e^{-i\beta /\left( {1/2+i\beta
_k } \right)}} ,
\end{equation}
such that
\begin{equation}
\label{eq159}
\Xi \left( {\beta ,\chi ^\ast } \right)\Xi \left( {-\beta ,\chi ^\ast }
\right)=\xi ^2\left( {\frac{1}{2},\chi ^\ast }
\right)\prod\limits_{k=1}^\infty {\left( {1-\frac{\beta ^2}{\beta _k^2 }}
\right)} .
\end{equation}
Let
\begin{equation}
\label{eq160}
\rho _k^\ast =Re\left( {\rho _k } \right)+i|Im\left( {\rho _k } \right)|
\end{equation}
such that $1-\rho _k $ are the nontrivial zeros for the
Dirichlet $L$--functions.

By \textit{Lemma \ref{L8}} and Eq.~(\ref{eq160}), we have
\begin{equation}
\label{eq161}
\begin{array}{l}
 \xi \left( {s,\chi ^\ast } \right)\\
=\xi \left( {0,\chi ^\ast }
\right)e^{B\left( {\chi ^\ast } \right)s}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s}{\rho _k^\ast}} \right)} e^{s/\rho _k^\ast } \\
 =\xi \left( {\frac{1}{2},\chi ^\ast } \right)e^{\left( {s-1/2}
\right)B\left( {\chi ^\ast } \right)}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s-\frac{1}{2}}{\rho _k^\ast -\frac{1}{2}}} \right)} e^{\left( {s-1/2}
\right)/\rho _k^\ast } \\
 \end{array}
\end{equation}
and
\begin{equation}
\label{eq162}
\begin{array}{l}
 \xi \left( {s,\chi ^\ast } \right)\\
=\xi \left( {0,\chi ^\ast }
\right)e^{B\left( {\chi ^\ast } \right)s}\prod\limits_{k=1}^\infty {\left(
{1-\frac{s}{1-\rho _k^\ast}} \right)} e^{s/\left( {1-\rho_k^\ast} \right)} \\
 =\xi \left( {\frac{1}{2},\chi ^\ast } \right)e^{\left( {s-1/2}
\right)B\left( {\chi ^\ast } \right)}\prod\limits_{k=1}^\infty {\left(
{1+\frac{s-\frac{1}{2}}{\rho _k^\ast -\frac{1}{2}}} \right)} e^{\left( {s-1/2}
\right)/\left( {1-\rho _k^\ast} \right)} \\
 \end{array}
\end{equation}
such that
\begin{equation}
\label{eq163}
\begin{array}{l}
 \xi ^2\left( {s,\chi ^\ast } \right) \\
 =\xi ^2\left( {0,\chi ^\ast } \right)e^{2sB\left( {\chi ^\ast }
\right)}\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{\rho _k^\ast}} \right)}
\prod\limits_{k=1}^\infty {\left( {1-\frac{s}{1-\rho _k^\ast}} \right)}
e^{\frac{s}{\rho _k^\ast \left( {1- _k^\ast} \right)}} \\
 =\xi ^2\left( {\frac{1}{2},\chi ^\ast } \right)e^{2\left( {s-1/2}
\right)B\left( {\chi ^\ast } \right)}\prod\limits_{k=1}^\infty {\left(
{1-\left( {\frac{s-\frac{1}{2}}{\rho _k^\ast-\frac{1}{2}}} \right)^2}
\right)e^{\frac{s-1/2}{\rho _k^\ast \left( {1-\rho _k^\ast} \right)}}} . \\
 \end{array}
\end{equation}
\end{remark}

According to \textit{Lemma \ref{L11}} and \textit{Conjecture \ref{C1}}, we set up the
following result.

\begin{corollary}
\label{Co1}

Let $\chi \left( n \right)$ be a Dirichlet character (mod $q>1)$. Suppose
that $N\left( {T,\chi } \right)$ denotes the number of zeros of the
Dirichlet $L$--function~(\ref{eq1}) for $Re\left( s \right)=1/2$, $T>2$ and $0\le
Im\left( s \right)\le T$ and $\widetilde{N}\left( {T,\chi } \right)$
denotes the number of zeros of the Dirichlet $L$--function~(\ref{eq1}) for
$0<Re\left( s \right)<1$, $T>2$ and $0\le Im\left( s \right)\le T$. Then we
have
\begin{equation}
\label{eq164}
N\left( {T,\chi } \right)=\widetilde{N}\left( {T,\chi } \right).
\end{equation}
\end{corollary}

\begin{proof}
From the obtained results in Section \ref{sec:3}, we find the followings:

When $\chi =\chi _0$ and $q=1$, Eq.~(\ref{eq164}) is true (see \cite{8,9} for further details), where
\[
L\left(
{s,\chi } \right)=\zeta \left( s \right).
\]

When $\chi =\chi _0$ and $q>1$, we have

\begin{equation}
\label{eqq165}
N\left( {T,\chi _0 } \right)=\widetilde{N}\left( {T,\chi _0 } \right),
\end{equation}
where $L\left( {s,\chi } \right)=L\left( {s,\chi _0 } \right)$.

When $\chi =\chi ^\ast$ and $\chi ^\ast \left( {-1} \right)=1$, we conclude that

\begin{equation}
\label{eq166}
N\left( {T,\chi ^\ast } \right)=\widetilde{N}\left( {T,\chi ^\ast }
\right).
\end{equation}
where $L\left( {s,\chi } \right)=L\left( {s,\chi ^\ast } \right)=\left(
{q/\pi } \right)^{-\frac{s}{2}}\xi \left( {s,\chi ^\ast } \right)/\Gamma
\left( {s/2} \right)$.

When $\chi =\chi ^\ast$ and $\chi ^\ast \left( {-1} \right)=-1$, we arrive at

\begin{equation}
\label{eq167}
N\left( {T,\chi ^\ast } \right)=\widetilde{N}\left( {T,\chi ^\ast }
\right),
\end{equation}
where $L\left( {s,\chi } \right)=L\left( {s,\chi ^\ast } \right)=\left(
{q/\pi } \right)^{-\frac{s+1}{2}}\xi \left( {s,\chi } \right)/\Gamma \left(
{s/2+1/2} \right)$.

Thus, we finish the proof.
\end{proof}

\subsection{The truth of the Goldbach's conjecture}

We now give new representations for the Wang theorems and present the applications of the Hardy--Littlewood theorems
to the Goldbach's conjecture because
\textit{Conjecture \ref{C1}} is proved and true.

\begin{theorem}
\label{T6}
(The weak Goldbach's conjecture)

Every odd number $m>5$ is the sum of three primes.
\end{theorem}

\begin{proof}
By \textit{Lemma \ref{L13}}, we have show the required result since
\textit{Conjecture \ref{C1}} is true.
\end{proof}

\begin{theorem}
\label{T7}  (Wang Theorem I)

$\left( {1,4} \right)$ is valid, where $\delta _2 \ge 3.237/2.237$.
\end{theorem}

\begin{proof}
By using \textit{Lemma \ref{L14}} and considering the fact \textit{Conjecture \ref{C1}} is
proved and true, we get the result.
\end{proof}

\begin{theorem}
\label{T8}

(Wang Theorem II)

$\left( {1,3} \right)$ is valid, where $\delta _1 \ge 2.475/1.475$.
\end{theorem}

\begin{proof}
Similarly, by using \textit{Lemma \ref{L15}}, we have the result because
\textit{Conjecture \ref{C1}} is true.
\end{proof}

\begin{theorem}
\label{T9}
(The strong Goldbach's conjecture)

Every even number $m>2$ is sums of two primes.
\end{theorem}

\begin{proof}
By \textit{Lemma \ref{L16}}, \textit{Theorem \ref{T9}} is true because of true of
\textit{Conjecture \ref{C1}}.
\end{proof}

\begin{remark}
By the work of
Deshouillers and coauthors~\cite{15}, we have that that the weak Goldbach's conjecture
is true because \textit{Conjecture \ref{C1}} is valid. By the work of Granville~\cite{18},
we also see that the strong Goldbach's conjecture is true
because \textit{Conjecture \ref{C1}} is valid. Thus, the Goldbach's conjecture is true.
\end{remark}

\section{Conclusion} {\label{sec:5}}

The present paper has proved that \textit{Conjecture \ref{C1}}
is true with the variant of the product for the entire function related to
the Dirichlet $L$--functions. We have presented the applications of it to the
Wang theorems for the Goldbach's problems.
By using the Hardy--Littlewood theorems, we have shown that every
odd number $m>5$ is the sum of three primes
and that every even number $m>2$ is sums of two primes.
The obtained
result is proposed to solve the mathematical problems under the assumption of the
truth of \textit{Conjecture \ref{C1}}.

\begin{acknowledgements}
This work is supported by the Yue-Qi Scholar of the China University of Mining and
Technology (No. 102504180004).
\end{acknowledgements}

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